wtf wrote:A set is infinite if it may be bijected with a proper subset of itself. That's the working definition.

How would you describe an infinite set of oranges using that definition of infinite?

Good question. I came up with two separate answers.

a) It's a theorem of set theory that every infinite set contains a countably infinite subset so it's no loss of generality to simply assume your set of oranges is countably infinite. If the set is uncountable we can adapt the same idea. So we label the oranges 0, 1, 2, 3, 4, ... We can do that since they're countable, which means there's a bijection between the naturals and the oranges. So we can number each orange by the natural number that maps to it in the bijection.

Now the entire set of oranges is in bijection with the set of even-numbered oranges, by the usual mapping n => 2n. Since the set of oranges can be bijected with a proper subset of itself, it's an infinite set of oranges.

b) Set theory as currently understood is purely about mathematical sets, the sets of ZFC or some similar axiom system. In ZFC, everything is a set. We start with the empty set, and the set containing the empty set, and the set containing those two, and so forth, and the the powersets and unions of all those sets, and so forth.

So in math, there is no set of oranges. If you have two oranges, I do NOT CLAIM that there is a set containing the two oranges. I do not personally believe in set theory outside of the pure sets of mathematics! That's essentially a formalist position. A formalist is a philosopher who maintains that math is simply about the formal manipulation of meaningless symbols according to arbitrary rules. It means NOTHING.

So there is no set of oranges. There are no sets of anything, other than the empty set and all the other sets that can be built from it via the axioms.

barbarianhorde wrote: if you look at the ways in which things are intertwined with each other, how many of them are there, because as soon as you understand something about them you intertwine with the web of intertwined ways of relating and then you see true infinity. But then you also see that infinity is what the world must become when it is held up by one perspective.

If you're saying what I think you're saying (bolded part), then you're hitting the nail on the head for the point I'm ultimately driving at which is there is no way to make an observation without affecting the thing being observed and, per Goethe, observation includes deduction. So, the fundamental of whatever we behold invariably will be perceived as infinite due to the infinite regression involved by affecting the thing being beheld, but that doesn't mean there is an existent infinity, but it simply means the subject and object are the same thing. So I would consider infinity to be proof of unity (the camera observing its own monitor) since the only alternative is to concede infinity exists as a completed incompletion, which is too nonsensical to get my head around and if we open the door to nonsense, how will we know where to draw the line.

Yes, thats what I meant but damn thats nicely phrased. So an object is infinitely grounded in itself. A set that describes a function such as a rational number sequence ids infinite in its potential reflection of itself on itself, where each reflection produces another integer, but there is not any infinitude of integers given unless that set is taken as the vessel. So the infinity opt the set is always infinity+1, the infinity of the number brings along the notion of the set. Which already shows its is not really infinite in capacity.

Damn, I dont now everyone can follow my thought here. Capacity as different from potential... well, like the capacity of a hose tied to an opened hydrant, and the potential of the closed hydrant and the rolled up hose.

Yes, a number only acquires a capacity to mean anything under certain circumstances, such as existence. lol.

Pi is one of these numbers. An irrational number. That speaks volumes. Its infinity is not a neat row or axial system, but more like snow on a tv. It is a better infinity if you want to come close to existence.

What I'm really driving at is the thing being perceived as infinite is really part of the one doing the observing and the perception of infinity is proof of that. If we are made of spacetime fabric stuff and we start inspecting the fundamentals of existence, then essentially what we are doing is looking at our own inner workings and self-inspection results in infinite regression, and self-inspection includes deductive means to peer inside which results in notions like infinite causality which should really mean lack of causality because the whole thing is one continuous thing giving rise to time itself as an emergent property rather than being subject to some objective time concept that would invariably have to be infinite.

When I try to picture an infinite plane in my mind, it can only curve back on itself because I'm trying to grasp the full extents of it as one thing and when I do that, it turns concave until it eventually joins with where my mind is calling the center. An infinite plane that extends forever without end isn't something I can imagine. I can fool myself into believing I can, but I'm lying if I claim such ability; the best I can do is make the edges fuzzy and call that infinite (that's cheating). But if I REALLY make a plane that doesn't end, then there is no other place to go than where it started. The only way to exist without also having beginning nor end is to be a loop (and why the wedding ring is a symbol of eternity and also a symbol of unity).

I think there is a way we can work with notions of infinity by working with the inevitable ramifications without actually conceptualizing infinity as a thing, but even that doesn't pan-out in practice. For instance the Thompson Lamp where the switch is turned on after 1 min and off after 1/2 min and on after 1/4 min and so on. Eventually the speed of the switch will exceed the speed of light, so whatever state the lamp was in before crossing the velocity threshold (probably on) will be the final state of the lamp since the switch would be moving too fast for electrons to react. So we can say in our heads that the final state of the lamp should be half-on and half-off, but it can't work in practice due to our finite universe. Even infinite velocity doesn't make sense since speed can't be faster than instant, which is c. Otherwise things could arrive before they left, and not only that, but arrive infinitely sooner than they left (whatever that means).

I think this is attained in the mirror loop metaphor for the set, where the set is a thing which is infinite inside but has no infinite capacity to change things, which would be infinite existence, which would mean infinite divisibility of meaning.

A mirror can only reflect images larger than its wavelength, so it's an illusion of infinity.

Conceptualizations of infinity are like that dream I had as a kid where a cat had its head in its own mouth (back in the days that I didn't realize bicycle spokes held the bike up and trees didn't make the wind blow); we probably can't admit to ourselves infinity is absurd because at such an old age, we shouldn't be that silly anymore, so it's denial. Compounding that, people really want infinity to exist since it's a good substitute for god and answers so many questions. The incentive to cheat (not be scientifically objective/unbiased) is high. And there is no proof or even good reason to believe an unbounded thing can be beheld by either our hands or our minds or even be said to exist, much like zero, which is also the bounded unbounded thing: limitless nothingness in a tidy package. Obviously we can think about "nothing" as the absence of something, but that's not nothing. I don't think anyone can truly think about nothing because there's nothing there to focus on, conceptualize, and observe.

Im compelled by this image of the trees that cause the wind I must say. Thats pretty damn cool. Yes, as kids we clearly have a lot more touch with the contradictions that are thrown at us, the way things are set against each other.

Out of the blue, it reminds me of my first solar eclipse when I was just a 4 year old kid walking home with my friend from getting some candy, I don't know why I was allowed, it was the 80s, and it suddenly got dark. It was a partial eclipse and no one was paying attention (it was the 80s) but for a moment I had the distinct sensation of "well that was it folks!". Later if you know what an eclipse is, it loses most of its capacity. Unless you're not in a horde of morons (humans) but in a field where suddenly every being is holding its breath, and you realize what you thought was silence was actually deafening noise.

Yes we had an eclipse last year I think (the one Trump looked at). I can't imagine what I would have thought about it if I didn't already know what it is.

wtf wrote:A set is infinite if it may be bijected with a proper subset of itself. That's the working definition.

How would you describe an infinite set of oranges using that definition of infinite?

Good question. I came up with two separate answers.

a) It's a theorem of set theory that every infinite set contains a countably infinite subset so it's no loss of generality to simply assume your set of oranges is countably infinite. If the set is uncountable we can adapt the same idea. So we label the oranges 0, 1, 2, 3, 4, ... We can do that since they're countable, which means there's a bijection between the naturals and the oranges. So we can number each orange by the natural number that maps to it in the bijection.

Now the entire set of oranges is in bijection with the set of even-numbered oranges, by the usual mapping n => 2n. Since the set of oranges can be bijected with a proper subset of itself, it's an infinite set of oranges.

b) Set theory as currently understood is purely about mathematical sets, the sets of ZFC or some similar axiom system. In ZFC, everything is a set. We start with the empty set, and the set containing the empty set, and the set containing those two, and so forth, and the the powersets and unions of all those sets, and so forth.

So in math, there is no set of oranges. If you have two oranges, I do NOT CLAIM that there is a set containing the two oranges. I do not personally believe in set theory outside of the pure sets of mathematics! That's essentially a formalist position. A formalist is a philosopher who maintains that math is simply about the formal manipulation of meaningless symbols according to arbitrary rules. It means NOTHING.

So there is no set of oranges. There are no sets of anything, other than the empty set and all the other sets that can be built from it via the axioms.

Either of those float your boat?

I'm not sure how definition b. describes an infinite set of oranges and how is definition a. any different than simply saying the number of oranges is unlimited/unbounded?

How would you describe infinite space? Would you say the number of sq inches can correspond to the number of sq feet? Again, how is that different from simply saying space is without end? Bijection couldn't be possible if sets had ends.

Wtf sorry for reading you wrong in that case! I didn't look it up by the way but talked to my math teacher, an old friend. I had dinner at his house and we discussed infinity.Then I went to corroborate some stuff online.

So since it seems we are somewhat in agreement again (I shift from side to side) let me ask you this, what is a subset of the irrationals that can be bijected with it?

Serendipper --- yes I definitely agree with that. I meant this means that whatever infinity is it is "inside" of a circuit, such as an observer hooked into an observation, the two together forming a world in a way, as a kind of closed system. At least closed for escaping. Stuff can still come in.

Such a feedbacking system is by definition infinite from the inside.

So a thing which is observed is part of the observing system and becomes hooked up to infinity. Lol. I need coffee.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

Serendipper wrote:I'm not sure how definition b. describes an infinite set of oranges and how is definition a. any different than simply saying the number of oranges is unlimited/unbounded?

In (b) I've taken the position that even finite "sets" of oranges don't exist, if by set we mean a mathematical set. And surely since there are only 10^78 hydrogen atoms in the observable universe, there can't be infinitely many oranges, whether contained in a set or not. So your question seems ill-founded.

What on earth do you mean by conceptualizing infinitely many oranges? What are they made of?

Serendipper wrote:How would you describe infinite space? Would you say the number of sq inches can correspond to the number of sq feet? Again, how is that different from simply saying space is without end? Bijection couldn't be possible if sets had ends.

If space is truly infinite, then the number of square (or cubic, or hypercubic) inches is the same as the number of square feet. Surely this is obvious. They're both infinite, and of the same cardinality.

barbarianhorde wrote:Wtf sorry for reading you wrong in that case! I didn't look it up by the way but talked to my math teacher, an old friend. I had dinner at his house and we discussed infinity.Then I went to corroborate some stuff online.

So since it seems we are somewhat in agreement again (I shift from side to side) let me ask you this, what is a subset of the irrationals that can be bijected with it?

The entire real line is in bijection with the unit interval. The bijections in each direction are the tangent and arctangent.

Another way to see something similar is the function f(x) = 1/x. That maps the open unit interval (0,1) to the entire positive real line.

ther way to see something similar is the function f(x) = 1/x. That maps the open unit interval (0,1) to the entire positive real line.

The real line is the real number line. The set of real numbers. Of course if you mean to exclude the rationals, the 1/x example still works since 1/x is rational if and only if x is.

So 1/x bijectively maps the set of irrationals in the unit interval (0,1) to the set of positive irrationals. And the irrationals in (0,1) are a proper subset of the set of positive irrationals.

You mean to include ignite between 1 and 0 - yes but thats the same as why naturals and rationals are the same class. what I try to figure out is how you can show one set (irrationals) is infinite by mapping a greater set (reals) onto it.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

barbarianhorde wrote:You mean to include ignite between 1 and 0 - yes but thats the same as why naturals and rationals are the same class.

No, completely different proof and idea. Well related, but not really the same.

barbarianhorde wrote: what I try to figure out is how you can show one set (irrationals) is infinite by mapping a greater set (reals) onto it.

I did. The set of all positive irrationals can be bijected onto one if its proper subsets, namely the irrationals strictly between 0 and 1.

Are you confused about the reals versus the irrationals? The reals include both the irrationals and the rationals. You can use the same proof idea for the reals or the irrationals. If you only care about the irrationals, you need to exclude the rationals.

Please tell me which part of this isn't clear. It's clear in my mind so perhaps I'm not understanding your question.

We show the reals are infinite by mapping them onto a proper subset.

We show the irrationals are infinite by mapping them onto a proper subset.

You could map the reals onto the irrationals bijectively, but it's a bit tricky and not worth the trouble.

Remember to show a set is infinite, I only have to biject it to SOME proper subset of itself. I don't have to biject it to any particular proper subset.

I do in fact know how to biject the reals to the irrationals, but it's a tricky construction and not worth going into detail about unless you want me to.

barbarianhorde wrote:This bijecting the reals to the irrationals is indeed what I was inquiring about, since what you said earlier about the real line hinges on it.

No that is not true, and it's a point you seem unclear on. Please take a moment to engage with this point, it's important.

A set is infinite if it can be bijected to at least one of its proper subsets. So the naturals are infinite because they can be bijected to the even naturals, or the odd naturals, or the primes, or (as Galileo noted in 1638) the perfect squares.

Likewise the reals are infinite because (0,1) is a proper subset and the tan/arctan functions biject the reals to (0,1). Or if you haven't taken trigonometry, you can biject (0,1) to the set of positive reals via f(x) = 1/x.

Please I request that you spend some time to understand this point.

Now, bijecting the reals to the irrationals is a curiosity. I don't need it to show the reals are infinite, the (0,1) examples already do that. But bijecting the reals to the irrationals is an interesting exercise, and shows how in general to get rid of a countable set within an uncountable one without altering the cardinality of the uncountable set.

So, here's a function that maps the reals to the irrationals.

First, the rationals are countable so they may be placed into an order like this: \(q_1, q_2, q_3, \dots\)

Now we need to choose any countable sequence of irrationals. It doesn't matter which one we choose, but for definiteness let's pick the sequence \(\pi, 2 \pi, 3 \pi, 4 pi, \dots\)

We define our function \(f(x)\) as follows. If \(x\) is rational, it's one of the \(q_n\)'s.We map each rational \(q_n\) to \(2 n \pi\). That is, we map \(q_1\) to \(2 \pi\), \(q_2\) to \(4 \pi\), and so forth.

If \(x\) is irrational and one of the \(n \pi\)'s, we map it to \((2 n - 1) \pi\). For example \(\pi\) goes to \(\pi\), \(2 \pi\) goes to \(3 \pi\), etc.

Finally, if \(x\) is anything else -- that is, if it's irrational and not one of the \(n \pi\)'s -- we map it to itself.

If you think this through (and I don't claim that's easy, this takes some work), you will see that we have a bijection between the reals and the irrationals.

This proof is due to Cantor. He used the irrational sequence \(\frac{\sqrt 2}{2^n}\) in order to show a bijection between the unit interval of reals and the irrationals in the unit interval.

Again, please note that showing the reals are an infinite set does not depend on this somewhat complicated example. We know the reals are infinite because we may biject them to (0,1), a proper subset of the reals. But if you ever need to show that there's a bijection between an uncountable set and that same set minus some countable set, this is the construction to use.

barbarianhorde wrote:Ah, this is the sort of reply I was hoping for. Yes, I will take some time. Thanks wtf.

You're very welcome. It's such a great construction and I enjoyed reviewing it myself.

I would feel better if I understood why you think it's important, because of the reasons I already mentioned ... that it's the (0,1) example that shows the reals (or the irrationals) are infinite, and the real -> irrational bijection is just a curiosity, although a nice one. But if you're happy I'm happy, and I'll stand by for questions. I'll be off the air the rest of the day but I'll be back tomorrow.

I really like it when abstract things are worked out to the concrete detail. It always, always always turns out to be valuable. Because it takes real effort from real minds.

I was pushing this topic a bit because I had been waiting for a good reason to get somewhat deeper into math. That worked very well, I now also have a better perspective on Russell, because I looked down on him pretty dramatically but I never knew of his type theory. I think that may be a bit underestimated, so far. I think it may become important in the future.

And yes I understand the proof now! I wish I was good enough at math to compare this to type theory. Still, I can begin to work at it now.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

barbarianhorde wrote:I really like it when abstract things are worked out to the concrete detail. It always, always always turns out to be valuable. Because it takes real effort from real minds.

Yes I agree. And it's cool that Cantor himself came up with this proof. It also serves as the standard technique for getting rid of countably many pesky problems. For example in Cantor's diagonal argument, we are listing the decimal representations of real numbers, but some real numbers have two distinct representations. For example .5 = .49999... So when we form the antidiagonal, how do we know that even if it's not on the list, its OTHER representation is?

The way this is usually handled is that we vigorously wave our hands (that part is essential) and say, "There are only countably many such pesky dual-representations so it doesn't matter." A thoughtful person might respond, "How do you know it doesn't matter?" Our example shows how to prove that the dual representations don't matter.

barbarianhorde wrote:I was pushing this topic a bit because I had been waiting for a good reason to get somewhat deeper into math. That worked very well, I now also have a better perspective on Russell, because I looked down on him pretty dramatically

I don't think any of us are in a position to do that! He was a great man in multiple endeavors. I don't know much about him, but why do you dislike him? Is it the pipe?

barbarianhorde wrote: but I never knew of his type theory.

How does that change your estimation of Russell?

barbarianhorde wrote: I think that may be a bit underestimated, so far. I think it may become important in the future.

There is no question that type theory is making a comeback as a potential foundation for math. The drive in this direction is from automated checking of proofs. There's a big movement called homotopy type theory (HOTT) that's the buzzphrase and Wiki page to know.

One thing I know from type-theoretic approaches to foundations is that they are associated with denial of the law of the excluded middle. I know very little about this entire area. It makes sense that this idea would gain currency in our age of computers; since there are many sets of natural numbers such that neither they nor their complement are computable. So neither is "true" in a world where truth is whatever can be determined by an algorithm.

The mathematical philosophy of all this goes under the name of intuitionism (if it's classic 1930's Brouwer type) or neo-intuitionism (if it's more recent).

h

barbarianhorde wrote:And yes I understand the proof now!

Yay! Me too! I had a vague idea about it before and now it's beautifully simple.

barbarianhorde wrote:I wish I was good enough at math to compare this to type theory. Still, I can begin to work at it now.

What leads you to your interest in type theory? Just curious. Well if HOTT eventually gains majority mindshare, I will think of that as Brouwer's revenge. There is another alternative foundation that's made huge inroads into much of modern research-level math, Category theory. Attitudes toward foundations are a matter of historical contingency.

Serendipper wrote:I'm not sure how definition b. describes an infinite set of oranges and how is definition a. any different than simply saying the number of oranges is unlimited/unbounded?

In (b) I've taken the position that even finite "sets" of oranges don't exist, if by set we mean a mathematical set. And surely since there are only 10^78 hydrogen atoms in the observable universe, there can't be infinitely many oranges, whether contained in a set or not. So your question seems ill-founded.

What on earth do you mean by conceptualizing infinitely many oranges? What are they made of?

I agree. Infinity can't exist.

Serendipper wrote:How would you describe infinite space? Would you say the number of sq inches can correspond to the number of sq feet? Again, how is that different from simply saying space is without end? Bijection couldn't be possible if sets had ends.

If space is truly infinite, then the number of square (or cubic, or hypercubic) inches is the same as the number of square feet. Surely this is obvious. They're both infinite, and of the same cardinality.

So you concede then that the infinite is boundless? How else could inch cubes and feet cubes correspond? (Sorry, I meant to say cubic feet and not square feet)

What I want to know is if there is any way sets can biject without being boundless.

Serendipper wrote:So you concede then that the infinite is boundless? How else could inch cubes and feet cubes correspond?

I responded to your own hypothetical about infinite space. Your response is disingenuous. You asked, IF space is infinite, is the number of cubic inches equal to the number of cubic feet. You posed a hypothetical. Jeez man.

Serendipper wrote:What I want to know is if there is any way sets can biject without being boundless.

Yes, the unit interval [0,1] bijects to the interval [0,2]. Both intervals are bounded.

This was 240 years before the advent of set theory. Bijection is a perfectly sensible notion even without a theory of infinite sets. The fingers on your hand are in bijection with the fingers on your glove.

wtf wrote:you can biject (0,1) to the set of positive reals via f(x) = 1/x.

I don't need it to show the reals are infinite, the (0,1) examples already do that.

How do you show infinite f(x) without having to first prove there are infinite x?

I don't understand the question. Do you deny that the unit interval (0,1) is in bijection with the positive reals via the map f(x) = 1/x? This is a fact familiar to high school students.

The chain of logic is as follows.

* I note that (0,1) is a proper subset of the positive reals.

* I note that (0,1) is in bijection with the positive reals via the map 1/x. If you doubt that you need to review your high school math.

* Since the set of positive reals are in bijection with one of its proper subsets. the positive reals are infinite by definition.

* I can do the same thing for the entire set of reals using the tangent/arctangent, but then you needed to have taken high school trig. The 1/x example has a more modest mathematical prerequisite so it's preferred for this conversation.